Multiple choice

From the top of a light house the angles of depression of two ships on the opposite sides of it are observed to be $\alpha$ and $\beta$. If the height of the light house be h meters and the line joining the ships passes through the foot of the light house, the distance between the ships is

  1. $\dfrac {h(cot\alpha+cot\beta)}{cot\alpha . cot \beta}$
  2. $\dfrac {h(tan\alpha+tan \beta)}{tan\alpha . tan\beta}$
  3. $h(tan\alpha + tan \beta)$
  4. $\dfrac {htan\alpha . tan \beta}{tan \alpha . tan\beta}$
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B Correct answer
Explanation

Let the distance from the foot of the lighthouse to the ships be x and y. Then tan(alpha) = h/x and tan(beta) = h/y. The total distance is x + y = h/tan(alpha) + h/tan(beta) = h(cot(alpha) + cot(beta)). This is equivalent to h(tan(alpha) + tan(beta)) / (tan(alpha) * tan(beta)).

AI explanation

Let h be the height of the lighthouse, and let the distances of the two ships from the foot be d1 and d2. Using the trigonometric ratios, cot(alpha) = d1/h and cot(beta) = d2/h, so d1 = h*cot(alpha) and d2 = h*cot(beta). The total distance between the ships is d1 + d2, which equals h*(cot(alpha) + cot(beta)). By converting cotangents to tangents using the identity cot(theta) = 1/tan(theta), this expression becomes h*((tan(beta) + tan(alpha))/(tan(alpha)tan(beta))), resulting in (h(tan(alpha) + tan(beta)))/(tan(alpha)*tan(beta)).

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